# How do you sketch the angle whose terminal side in standard position passes through (-3,1) and how do you find sin and cos?

Aug 18, 2017

#### Explanation:

To sketch the angle in standard position,

One side of the angle is the positive $x$ axis (the right side of the horizontal axis). That is the initial side.

The terminal side has one end at the origin (the point $\left(0 , 0\right)$, also the intersection of the two axes) and goes through the point $\left(- 3 , 1\right)$. So locate the point $\left(- 3 , 1\right)$. Starting at the origin and count $3$ to the left and up $1$. That will get you to the point $\left(- 3 , 1\right)$. Put a dot there.
Now draw a line from the origin through the point $\left(- 3 , 1\right)$. Your sketch should look a lot like this:

If we knew how the angle was made (which direction and how many times around the circle), we would show that also.

Give the angle a name. I will use $\theta$ (that is the Greek letter "theta")

Memorize this

If the point $\left(a , b\right)$ lies on the terminal side of an angle in standard position, then let $r = \sqrt{{a}^{2} + {b}^{2}}$

The angle has sine $\frac{b}{r}$ and it has cosine $\frac{a}{r}$

For this question

We have $\left(a , b\right) = \left(- 3 , 1\right)$, so

$r = \sqrt{{\left(- 3\right)}^{2} + {\left(1\right)}^{2}} = \sqrt{9 + 1} = \sqrt{10}$.

So the sine of $\theta$ is

$\sin \left(\theta\right) = \frac{1}{\sqrt{10}}$

Many trigonometry teachers will insist that you write your answer with a rational number in the denominator. If this is something you have to do, multiply the answer by $\frac{\sqrt{10}}{\sqrt{10}}$ to look like this:

$\frac{1}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}} = \frac{1 \cdot \sqrt{10}}{\sqrt{10} \cdot \sqrt{10}} = \frac{\sqrt{10}}{10}$

$\sin \left(\theta\right) = \frac{\sqrt{10}}{10}$
The cosine of $\theta$ is $\frac{a}{r}$ so we have
$\cos \left(\theta\right) = \frac{- 3}{\sqrt{10}} = \frac{- 3}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}} = - \frac{3 \sqrt{10}}{10}$
$\cos \left(\theta\right) = - \frac{3 \sqrt{10}}{10}$